Let us explain why our approach works and how it differs from what was developed in [1]}, [2]}. To achieve this goal, we need to focus on the functions (REF ) and (REF ), the relation (REF ), and the fact that the function \(f_t(\cdot )\)  is non-negative, for each \(t\ge 0\) . The two aforementioned functions are extensions of functions introduced in [1]}, which can be recovered when \(\rho =0\)  once \(t>0\)  is fixed. In [2]} we showed that if it was possible to find such functions for a single value of \(t\) , then it would be possible to construct one-parameter families of such function, whose parameter is the first variable, or the time in physical terms. This fact is widely explored in Section  of the present work and we also made an in-depth investigation about the properties of these two families. In particular, for \(t>0\) , the family \(F_t\)  can be rewritten using the relation (REF ), whereas the relation (REF ) holds for all values of \(t\) . These two conditions are key ingredients for proving nearly all the main results of the present work and, in particular, Theorem REF .
